- Research
- Open Access
Blow-up properties for a degenerate parabolic system coupled via nonlinear boundary flux
- Si Xu^{1}Email author
- Received: 11 July 2015
- Accepted: 28 September 2015
- Published: 15 October 2015
Abstract
In this paper, we study the simultaneous and non-simultaneous blow-up problem for a system of two nonlinear diffusion equations in a bounded interval, coupled at the boundary in a nonlinear way. Under certain hypotheses on the initial data and parameters, we prove that non-simultaneous blow-up is possible. Moreover, we get some conditions on which simultaneous blow-up must occur, as well as the non-simultaneous blow-up conditions for every initial data. Furthermore, we get a result on the coexistence of both simultaneous and non-simultaneous blow-ups.
Keywords
- non-simultaneous blow-up
- blow-up rate
- nonlinear diffusion
- nonlinear boundary flux
MSC
- 5B33
- 35K65
- 35K55
1 Introduction and main results
Remark 1.1
The reaction-diffusion system (1.1)-(1.3) can be used to describe heat transfer in a mixed medium with absorptions and nonlinear boundary flux, and some chemical reaction processes with the slow diffusion phenomenon (see [1, 2]).
- (1)
\(\max\{\alpha, p_{1}\}>1\),
- (2)
\(\max\{\beta, q_{2}\}>1\),
- (3)
\(p_{2}q_{1}>(1-p_{1})(1-q_{2})\).
Recently, the simultaneous and non-simultaneous blow-up problems of parabolic systems have been widely considered by many authors [4–14]. For example, when \(m=n=1\), \(\lambda_{i}=0\), Pinasco and Rossi [6] considered the system of heat equations coupled via a nonlinear boundary flux with \(\Omega\subset R^{N}\) and found that u blows up at time T and v remains bounded up to time T for certain initial data if and only if \(p_{1}>1\) and \(p_{2}< p_{1}-1\).
When \(m=n=1\) and \(\lambda_{i}<0\), Zheng and Qiao [10] also got the sufficient and necessary conditions of non-simultaneous blow-up.
In this paper, by using a modification of methods in [4] and [10] we will focus on the simultaneous and non-simultaneous blow-up problems to (1.1)-(1.3), and we get our main results as follows.
Theorem 1.1
Theorem 1.2
When \(\lambda_{i}>0\) and \(\max\{\alpha, p_{1}\}>m\), if u blows up at time T and v remains bounded up to T, then (1.5) holds.
Theorem 1.3
When \(\lambda_{i}<0\), if \(p_{1}>m\) and \(2p_{1}-m>\max\{2p_{2}+1, \alpha\}\), then there exists initial data \((u_{0}, v_{0})\) such that u blows up at a finite time T, while v remains bounded up to T.
By interchanging the roles of u and v, we get the following results.
Corollary 1.4
- (i)if \(\max\{\beta,q_{2}\}>n\) andthen there exists initial data \((u_{0}, v_{0})\) such that v blows up at a finite time T, while u remains bounded up to T.$$ \max\{2q_{2}-n,\beta\}>2q_{1}+1, $$(1.6)
- (ii)
When \(\max\{\beta,q_{2}\}>n\), if v blows up at a finite time T and u remains bounded up to T, then (1.6) holds.
Corollary 1.5
When \(\lambda_{i}<0\), if \(q_{2}>n\) and \(2q_{2}-n>\max\{2q_{1}+1, \beta\}\), then there exists initial data \((u_{0}, v_{0})\) such that v blows up at a finite time T, while u remains bounded up to T.
Theorem 1.6
When \(\lambda_{i}>0\), if \(\max\{\alpha, p_{1}\}>m\) and \(2p_{2}+1\geq\max\{2p_{1}-m,\alpha\}\), then for any initial data \((u_{0}, v_{0})\) the solution \((u,v)\) to (1.1)-(1.3) blows up simultaneously at a finite time T.
Theorem 1.7
When \(\lambda_{i}>0\), if \(\max\{\alpha, p_{1}\}>m\) and (1.5) holds, then the set of initial data such that u blows up and v remains bounded is open in the \(L^{\infty}\) topology. If \(\max\{\beta, q_{2}\}>n\) and (1.6) holds, then the set of initial data such that v blows up and u remains bounded is open in \(L^{\infty}\) topology.
Next, inspired by [9, 12], we consider the coexistence of both simultaneous and non-simultaneous blow-ups, and we get the blow-up rate if simultaneous blow-ups occur.
Theorem 1.8
Finally, let us show that, under certain conditions, blow-up is always non-simultaneous.
Theorem 1.9
Theorem 1.10
When \(\lambda_{i}>0\), if \(\max\{\alpha, p_{1}\}>m\), \(\max\{\beta, q_{2}\}\leq1\) and (1.5) holds, then u blows up and v remains bounded for any initial data.
The rest of this paper is organized as follows. In the next section, we first get some blow-up results of scalar problems, and using them to prove Theorems 1.1-1.3. In Section 3, we consider the coexistence of both simultaneous and non-simultaneous blow-ups, Theorems 1.7, 1.8 are proved. In Section 4, we show that non-simultaneous blow-ups always occur, and we prove Theorems 1.9 and 1.10.
2 Non-simultaneous and simultaneous blow-up
When \(\lambda_{1}<0\), we know from [18], if \(p>l\) and \(q<2p-l\), that the solutions of (2.1) blow up for large initial data.
We give the blow-up rate of the solution of (2.1) in the following two lemmas.
Lemma 2.1
- (i)
When \(2p>q+l\), \(\sigma=\frac{1}{2p-l-1}\).
- (ii)
When \(2p< q+l\), \(\sigma=\frac{1}{q-1}\).
- (iii)
When \(2p=q+l\), \(\sigma=\frac{1}{q-1}=\frac{1}{2p-l-1}\).
Proof
Notice that if case (i) holds, by \(\max\{q, p\}>l\) and \(2p>q+l\), we get \(p>l\). If case (ii) holds, it follows from \(\max\{q, p\}>l\) and \(2p< q+l\) that \(q>l\). Thus, by a similar proof to that of Theorem 3.1 in [15], we can prove easily Lemma 2.1. We omit it here. □
Lemma 2.2
Proof
The proof is similar to that of Theorem 1.2 in [18]. We omit it here. □
Lemma 2.3
Proof
Lemma 2.4
Proof
Lemma 2.5
Proof
It is an immediate conclusion of Theorem 3.2(i) in [4]. □
Proof of Theorem 1.1
Since \(v\geq\delta>0\), for all \((x,t)\in[0,1]\times[0,T)\), u is a supersolution of \(\underline{u}\). Then u also blows up, and the blow-up time T is smaller than \(T'\). Thus we have \(T\leq T'\leq(\frac{C}{u_{0}(x)})^{1/\sigma}\). As \(\sigma>0\), we can choose the initial data \(u_{0}(x)\) large enough such that T is small.
- (i)
If \(2p_{1}>\alpha+m\), from (1.5) we have \(2p_{1}-m>2p_{2}+1\), thus \(s=p_{2}\sigma=\frac{p_{2}}{2p_{1}-m-1}<1/2\).
- (ii)
If \(2p_{1}<\alpha+m\), from (1.5) we have \(\alpha>2p_{2}+1\), also get \(s=\frac{p_{2}}{\alpha-1}<1/2\).
- (iii)If \(2p_{1}=\alpha+m\), from (1.5) we have \(2p_{1}-m=\alpha=2p_{2}+1\), and we get$$s=\frac{p_{2}}{\alpha-1}=\frac{p_{2}}{2p_{1}-m-1}< 1/2. $$
Proof of Theorem 1.2
- (i)
If \(2p_{1}>\alpha+m\), from \(s=\frac{p_{2}}{2p_{1}-m-1}<1/2\), we have \(2p_{1}-m>\max\{2p_{2}+1, \alpha\}\).
- (ii)
If \(2p_{1}<\alpha+m\), from \(s=\frac{p_{2}}{\alpha-1}<1/2\), we have \(\alpha>\max\{2p_{2}+1, 2p_{1}-m\}\).
- (iii)
If \(2p_{1}=\alpha+m\), from \(s=\frac{p_{2}}{\alpha-1}=\frac{p_{2}}{2p_{1}-m-1}<1/2\), we have \(2p_{1}-m=\alpha>2p_{2}+1\).
From the previous three cases, we have (1.5). □
3 Coexistence of simultaneous and non-simultaneous blow-ups
In this section, we consider the coexistence of both simultaneous and non-simultaneous blow-ups, and we prove Theorems 1.7 and 1.8.
Proof of Theorem 1.7
Let \((u, v)\) be a solution of (1.1)-(1.3) with initial data \((u_{0}, v_{0})\) such that u blows up at T while v remains bounded, that is, \(v\leq C\). We only need to find a \(L^{\infty}\)-neighborhood of \((u_{0}, v_{0})\) such that any solution \((\hat{u}, \hat{v})\) of (1.1)-(1.3) with initial data \((\hat{u_{0}},\hat{v_{0}})\) in this neighborhood maintains the property that u blows up while v remains bounded.
In fact, as the solutions of (1.1)-(1.3) are bounded up to time \(T-\varepsilon\), the continuity of solution respect to initial conditions holds up to \(T-\varepsilon\), which means for any \(\epsilon_{0}\), there is a \(\delta_{0}\), \(\|(\hat{u_{0}}, \hat{v_{0}})-(u_{0}, v_{0})\|_{\infty}<\delta_{0}\), such that \(\|(\hat{u}, \hat{v})-(u, v)\|_{\infty}<\epsilon_{0}\). When we let \(\epsilon_{0}=1\), there is a \(\delta_{0}\), such that \(\|(\hat{u}, \hat{v})-(u, v)\|_{\infty}<1\), so we get \(\|\hat{u}-u\|_{\infty}<1\) and \(\|\hat{v}-v\|_{\infty}<1\). Moreover, u becomes large at time \(T-\varepsilon\) and \(v\leq C\) up to \(T-\varepsilon\), so have found a neighborhood of \((u_{0},v_{0})\) in \(L^{\infty}\) such that, if \((\hat{u}, \hat{v})\) has initial data in such neighborhood, then û becomes large at time \(T-\varepsilon\) and \(\hat{v}\leq C+1\) up to \(T-\varepsilon\). The argument in the proof of Theorem 1.1 allows us to conclude that û blows up and v̂ remains bounded, if we consider time \(T-\varepsilon\) as the initial time. □
Proof of Theorem 1.8
Under our assumptions, from Theorem 1.1, we know that the set of \((u_{0}, v_{0})\) such that u blows up and v remains bounded is nonempty. From Corollary 1.4(i), we also know the set of initial data for v blowing up and u being bounded is nonempty. Moreover, Theorem 1.7 concludes that such sets are open. Clearly, the two open sets are disjoint. That is to say, there exist \((u_{0}, v_{0})\) such that u and v blow up simultaneously at a finite time T.
4 Non-simultaneous blow-ups always happen
In this section, we will show that under certain conditions non-simultaneous blow-ups always occur, and we will prove Theorems 1.9 and 1.10.
Proof of Theorem 1.9
Proof of Theorem 1.10
It is well known that under the assumptions, v cannot blow up without the help of u. So the blow-up time of v cannot be larger than that of u. u blows up at finite time T, by considering the solution at time \(T-\varepsilon\) as initial data, we may assume that the blow-up time is as small as desired. Following the same steps of the proof of Theorem 1.1, the conclusion follows. □
Declarations
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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